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Last updated on August 5th, 2025

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Derivative of 9lnx

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We use the derivative of 9ln(x), which is 9/x, to understand how the logarithmic function changes with respect to x. Derivatives are crucial in calculating profit or loss in real-world scenarios. We will now discuss the derivative of 9ln(x) in detail.

Derivative of 9lnx for Australian Students
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What is the Derivative of 9lnx?

The derivative of 9ln(x) is denoted as d/dx (9ln(x)) or (9ln(x))', and its value is 9/x. This derivative indicates how the function 9ln(x) changes in response to variations in x, highlighting its differentiability within its domain.

 

Key concepts include:

 

Logarithmic Function: ln(x) is the natural logarithm of x.

 

Constant Multiple Rule: Used for differentiating a constant multiplied by a function.

 

Reciprocal Function: The derivative of ln(x) is 1/x.

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Derivative of 9lnx Formula

The derivative of 9ln(x) is expressed as d/dx (9ln(x)) or (9ln(x))'. The formula used to differentiate 9ln(x) is: d/dx (9ln(x)) = 9/x

 

This formula is applicable for all x > 0, as ln(x) is only defined for positive x.

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Proofs of the Derivative of 9lnx

We can derive the derivative of 9ln(x) using proofs. To demonstrate this, we will apply differentiation rules and logarithmic identities. Several methods can be employed, such as:

 

  1. By First Principle
  2. Using Constant Multiple Rule
  3. Using Chain Rule

 

We will now show that differentiating 9ln(x) results in 9/x using the methods mentioned:

 

By First Principle

 

The derivative of 9ln(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

 

To find the derivative of 9ln(x) using the first principle, consider f(x) = 9ln(x). Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 9ln(x), we write f(x + h) = 9ln(x + h).

 

Substituting these into equation (1), f'(x) = limₕ→₀ [9ln(x + h) - 9ln(x)] / h = 9limₕ→₀ [ln(x + h) - ln(x)] / h = 9limₕ→₀ [ln((x + h)/x)] / h = 9limₕ→₀ ln[(1 + h/x)] / h

 

Using the limit property, limₕ→₀ ln(1 + u)/u = 1, f'(x) = 9 * 1/x Hence, f'(x) = 9/x, proved.

 

Using Constant Multiple Rule

 

To prove the differentiation of 9ln(x) using the constant multiple rule, Consider the constant 9 and the function ln(x). Using the rule: d/dx [c·f(x)] = c·f'(x)

 

Here, c = 9 and f(x) = ln(x), f'(x) = d/dx [ln(x)] = 1/x Thus, d/dx [9ln(x)] = 9 * 1/x = 9/x.

 

Using Chain Rule

 

The chain rule can also be used to differentiate 9ln(x). Consider y = 9ln(x) Let u = ln(x), then y = 9u

 

By chain rule: dy/dx = dy/du * du/dx dy/du = 9 and du/dx = 1/x

 

Therefore, dy/dx = 9 * 1/x = 9/x.

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Higher-Order Derivatives of 9lnx

When a function is differentiated multiple times, the resulting derivatives are higher-order derivatives. Higher-order derivatives can be complex, similar to understanding a car's speed (first derivative) and its acceleration (second derivative). Higher-order derivatives offer insights into functions like 9ln(x).

 

For the first derivative, we write f′(x), indicating the slope or rate of change at a point. The second derivative, f′′(x), is derived from the first derivative. The third derivative, f′′′(x), follows from the second derivative, continuing this pattern.

 

For the nth derivative of 9ln(x), we denote it as fⁿ(x), providing insights into the rate of change for each derivative order.

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Special Cases:

When x is 0, the derivative is undefined as ln(x) is not defined for non-positive x. When x is 1, the derivative of 9ln(x) = 9/1 = 9.

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Common Mistakes and How to Avoid Them in Derivatives of 9lnx

Mistakes often occur when differentiating 9ln(x). These can be avoided by understanding the correct procedures. Here are some common mistakes and solutions:

Mistake 1

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Not applying the constant multiple rule

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Students may forget to apply the constant multiple rule, leading to incorrect results. It's important to factor the constant 9 out before differentiating ln(x). Ensure each step is followed correctly and don't skip the application of the constant multiple rule.

Mistake 2

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Ignoring the domain of ln(x)

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Students might overlook that ln(x) is undefined for x ≤ 0. Always consider the domain of the function when differentiating, as this ensures the result is valid for the given inputs.

Mistake 3

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Incorrect use of the chain rule

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While differentiating composite functions like ln(ax), students might misapply the chain rule. For example, they may incorrectly differentiate ln(3x) without considering the inner function. Always use the chain rule correctly by identifying inner and outer functions and differentiating both.

Mistake 4

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Not simplifying the final expression

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A common mistake is leaving the derivative in a complex form without simplification. Simplifying the final result can help in understanding and verifying the correctness of the derivative.

Mistake 5

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Incorrect limit evaluation

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Errors often occur in evaluating limits when using the first principle. Always apply the correct limit rules and verify each step to ensure the result is accurate.

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Examples Using the Derivative of 9lnx

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Problem 1

Calculate the derivative of (9ln(x) · x²)

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Here, we have f(x) = 9ln(x) · x².

 

Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 9ln(x) and v = x².

 

Differentiate each term, u′= d/dx (9ln(x)) = 9/x v′= d/dx (x²) = 2x

 

Substituting into the given equation, f'(x) = (9/x) · x² + 9ln(x) · 2x

 

Simplify terms to get the final answer, f'(x) = 9x + 18xln(x)

 

Thus, the derivative of the specified function is 9x + 18xln(x).

Explanation

We find the derivative of the given function by dividing it into two parts. First, we find the derivative of each part and then combine them using the product rule to get the final result.

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Problem 2

A construction company uses the function y = 9ln(x) to model the pressure in a pipe at a distance x meters from the source. If x = 2 meters, measure the rate of change of pressure.

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We have y = 9ln(x) (pressure in the pipe)...(1)

 

Differentiate equation (1) dy/dx = 9/x

 

Given x = 2, substitute this into the derivative dy/dx = 9/2

 

Hence, the rate of change of pressure at x = 2 meters is 4.5.

Explanation

We calculate the rate of change at x = 2 meters as 4.5, indicating how the pressure changes with respect to distance from the source.

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Problem 3

Derive the second derivative of the function y = 9ln(x).

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First, find the first derivative, dy/dx = 9/x...(1)

 

Differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [9/x] = -9/x²

 

Therefore, the second derivative of the function y = 9ln(x) is -9/x².

Explanation

We use a step-by-step process, starting with the first derivative. We then differentiate 9/x to obtain the second derivative, simplifying to find the final answer.

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Problem 4

Prove: d/dx (ln²(x)) = 2ln(x)/x.

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Start using the chain rule: Consider y = ln²(x) = [ln(x)]²

 

Differentiate using the chain rule: dy/dx = 2ln(x) · d/dx [ln(x)]

 

Since the derivative of ln(x) is 1/x, dy/dx = 2ln(x)/x Hence proved.

Explanation

In this step-by-step process, we use the chain rule to differentiate the equation. We replace ln(x) with its derivative and simplify to derive the equation.

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Problem 5

Solve: d/dx (ln(x)/x)

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To differentiate the function, use the quotient rule: d/dx (ln(x)/x) = (d/dx (ln(x)) · x - ln(x) · d/dx(x))/x²

 

Substitute d/dx (ln(x)) = 1/x and d/dx (x) = 1 (1/x · x - ln(x) · 1)/x² = (1 - ln(x))/x²

 

Therefore, d/dx (ln(x)/x) = (1 - ln(x))/x²

Explanation

In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.

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FAQs on the Derivative of 9lnx

1.Find the derivative of 9ln(x).

Using the constant multiple rule, the derivative of 9ln(x) is 9/x.

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2.Can we use the derivative of 9ln(x) in real life?

Yes, derivatives of functions like 9ln(x) are used in real-life applications such as calculating growth rates in biology, economics, and engineering.

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3.Is it possible to take the derivative of 9ln(x) at x = 0?

No, x = 0 is a point where ln(x) is undefined, so it is impossible to take the derivative at this point.

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4.What rule is used to differentiate ln(x)/x?

We use the quotient rule to differentiate ln(x)/x, resulting in d/dx (ln(x)/x) = (1 - ln(x))/x².

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5.Are the derivatives of 9ln(x) and ln⁻¹(x) the same?

No, they are different. The derivative of 9ln(x) is 9/x, while the derivative of ln⁻¹(x) is -1/(x² ln²(x)).

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Important Glossaries for the Derivative of 9lnx

  • Derivative: The derivative of a function measures how it changes in response to a change in x.

 

  • Logarithmic Function: A function involving the logarithm, typically represented as ln(x).

 

  • Constant Multiple Rule: A differentiation rule where a constant is factored out before differentiating a function.

 

  • Chain Rule: A rule for differentiating composite functions, involving the differentiation of both outer and inner functions.

 

  • Quotient Rule: A rule used for differentiating a function that is the quotient of two other functions.
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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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